Does my Lumia phone use an IP address when it accesses data networks?

Yes. When using data networks (cellular data or WLAN) to access for example Internet, your phone uses an IP address, either a dynamic IP address (default) received from the network or a static IP address specified in the phone.

At this time, there is no way to assign a static IP address to your Windows Phone Wi-Fi. As a possible workaround, you will find out the MAC address of your Windows Phone device and then set up a DHCP reservation at a particular IP for that MAC address. You can access your MAC address by going into Settings > About > More Info and see you MAC address listed.